Re: Monadic-Mathematics (a corrected post)


DoronShadmi wrote:
> > On Fri, 16 Dec 2005 18:29:44 EST, DoronShadmi
> > <shdmy-d@xxxxxxxxx>
> > wrote something.
> >
> >
> > Set theoretic definitions:
> >
> > A c= B =df Ax(x e A -> x e B)
> > "A is a subset of B."
> >
> > A c B =df A c= B & ~(B c= A).
> > "A is a proper subset of B."
> >
> >
> > Substituting the definition of c= we get from the
> > latter definition:
> >
> > A c B <-> Ax(x e A -> x e B) & ~Ax(x e B -> x e A).
> >
> >
> > Now you write:
> >
> > "Each one of {2,4,6,...} is also in
> > n {1,2,3,4,5,6,...} but not
> > vice versa."
> >
> > Hence:
> >
> > {2,4,6,...} c {1,2,3,4,5,6,...}.
> >
> > qed.
> >
> >
> > F.
> >
> > --
> > "I do tend to feel Hughes & Cresswell is a more
> > authoritative
> > source than you." (D. Ullrich)
>
> Hi D. Ullrich,
>
> You missed my point, because {2,4,6,...} is a proper subset of {1,2,3,4,5,6,...}
> iff |{2,4,6,...}| < |{1,2,3,4,5,6,...}| and since this is not the case, no logical tricks can ignore the fact that finite and non-finite collections of natural numbers are *not* in the same category, and you cannot expand the results that belong to finite sets, to non-finite sets.

If he missed your point, it's only because what you write is very
confusing to anyone using the standard definitions of mathematics. You
use the term "subset" like most people do, but then give it a funny
definition. G. Frege was merely pointing out that by any normal
definition of subset that {2, 4, 6, ...} is clearly a subset of the
naturals. You seem to be rejecting this definition, although I'm not
sure what your purpose for doing this is. In fact, this definition
seems rather problematic. For one thing, the only possible subsets of
an infinite set are sets of a stricly smaller cardinality. So, for
example, we can no longer conclude that the interval [0, 1] is a subset
of the reals. For the same reason, I'm not sure if we could say that
any infinite set is a subset of itself. Furthermore, say that A and B
are infinite subsets with the same caridinality such that A is a subset
of B and B is a subset of A using the normal definition of subset.
Usually, we would then conclude that A = B, but I don't see how we
would do this using your definition.

Perhaps all the issues I mentioned above are your reason for creating
some new set theory. If this is the case, however, I still don't
understand what you think is lacking with the standard way of defining
a subset.

Mike

.

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